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We consider an $n\times n$ linear system of ODEs with an irregularsingularity of Poincar\'e rank 1 at $z=\infty$, holomorphically depending onparameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. Theeigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $\Delta$contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is aresonant irregular singularity for $t\in \Delta$. We analyse the case when theleading matrix remains diagonalisable at $\Delta$. We discuss the existence offundamental matrix solutions, their asymptotics, Stokes phenomenon andmonodromy data as $t$ varies in the polydisc, and their limits for $t$ tendingto points of $\Delta$. When the deformation is isomonodromic away from$\Delta$, it is well known that a fundamental matrix solution has singularitiesat $\Delta$. When the system also has a Fuchsian singularity at $z=0$, we showunder minimal vanishing conditions on the residue matrix at $z=0$ thatisomonodromic deformations can be extended to the whole polydisc, including$\Delta$, in such a way that the fundamental matrix solutions and the constantmonodromy data are well defined in the whole polydisc. These data can becomputed just by considering the system at fixed $t=0$. Conversely, if the$t$-dependent system is isomonodromic in a small domain contained in thepolydisc not intersecting $\Delta$, if the entries of the Stokes matrices withindices corresponding to coalescing eigenvalues vanish, then we show that$\Delta$ is not a branching locus for the fundamental matrix solutions. Theimportance of these results for the analytic theory of Frobenius Manifolds isexplained. An application to Painlev\'e equations is discussed.

Isomonodromy Deformations at an Irregular Singularity with Coalescing Eigenvalues